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Davide Lombardo

Publications and source records attributed to Davide Lombardo.

At least 37 records · Page 2Linked to original sources
arXiv2023

Torsion bounds for a fixed abelian variety and varying number field

Let $A$ be an abelian variety defined over a number field $K$. For a finite extension $L/K$, the cardinality of the group $A(L)_{\operatorname{tors}}$ of torsion points in $A(L)$ can be bounded in terms of the degree $[L:K]$. We study the smallest real number $β_A$ such that for any finite extension $L/K$ and $\varepsilon>0$, we have $|A(L)_{\operatorname{tors}}| \leq C \cdot [L:K]^{β_A+\varepsilon}$, where the constant $C$ depends only on $A$ and $\varepsilon$ (and not $L$). Assuming the Mumford--Tate conjecture for $A$, we will show that $β_A$ agrees with the conjectured value of Hindry and Ratazzi. We also give a similar bound for the maximal order of a torsion point in $A(L)$.

math.NT↗
arXiv2023

Cohomology of groups acting on vector spaces over finite fields

Let ${\mathbf{F}}_q$ be the finite field with $q=p^m$ elements and $G$ be a subgroup of ${\rm{GL}}_n({\mathbf{F}}_q)$. A famous theorem of Nori published in 1987 states that there exists a (non-effective) constant $c(n)$, depending only on $n$, such that if $p>c(n)$ and $G$ acts semisimply on ${\mathbf{F}}_p^n$, then $H^1(G,{\mathbf{F}}_p^n)=0$. We solve the long-standing problem, also considered by Serre of giving an effective proof of Nori's Theorem. Our approach yields the optimal constant $c(n)=n+2$. We also prove a more general version of Nori's theorem, namely, that for all powers $q$ of $p$, if $G$ acts semisimply on ${\mathbf{F}}_q^n$ and $p>n+2$, then $H^1(G,{\mathbf{F}}_q^n)$ is trivial. We apply these results to refine a criterion, proved by Çiperiani and Stix, which gives sufficient conditions for an affirmative answer to a classical question posed by Cassels in the case of abelian varieties over number fields.

math.NT↗
arXiv2022

A family of quintic Thue equations via Skolem's $p$-adic method

In this semi-expository article we solve the diophantine equation $m^5+(4 \cdot 5^4 b^4)mn^4 - n^5=1$ for all integers $b \neq 0$. This gives an example of a family of quintic Thue equations that can be solved completely by using nothing more than Skolem's $p$-adic method. We also give a general introduction to Skolem's method from a modern perspective.

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arXiv2022

On the distribution of rational points on ramified covers of abelian varieties

We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields $k$ of characteristic zero. For example, given a ramified cover $π: X \to A$, where $A$ is an abelian variety over $k$ with a dense set of $k$-rational points, we prove that there is a finite-index coset $C \subset A(k)$ such that $π(X(k))$ is disjoint from $C$. Our results do not seem to be in the range of other methods available at present; they confirm predictions coming from Lang's conjectures on rational points, and also go in the direction of an issue raised by Serre regarding possible applications to the Inverse Galois Problem. Finally, the conclusions of our work may be seen as a sharp version of Hilbert's irreducibility theorem for abelian varieties.

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arXiv2021

Non-isogenous abelian varieties sharing the same division fields

Two abelian varieties $A_1, A_2$ over a number field $K$ are called strongly iso-Kummerian if the torsion fields $K(A_1[d])$ and $K(A_2[d])$ coincide for all $d \geq 1$. For all $g \geq 4$ we construct geometrically simple, strongly iso-Kummerian $g$-dimensional abelian varieties over number fields that are not geometrically isogenous. We also discuss related examples and put significant constraints on any further iso-Kummerian pair.

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arXiv2021

Some uniform bounds for elliptic curves over $\mathbb Q$

We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves $E/\mathbb Q$. We consider in particular the subgroup of scalars in the image of Galois, the first Galois cohomology group with values in the torsion of $E$, and the Kummer extensions generated by points of infinite order in $E(\mathbb Q)$.

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arXiv2021

Computing the geometric endomorphism ring of a genus 2 Jacobian

We describe an algorithm, based on the properties of the characteristic polynomials of Frobenius, to compute $\operatorname{End}_{\overline{K}}(A)$ when $A$ is the Jacobian of a nice genus-2 curve over a number field $K$. We use this algorithm to confirm that the description of the structure of the geometric endomorphism ring of $\operatorname{Jac}(C)$ given in the LMFDB ($L$-functions and modular forms database) is correct for all the genus 2 curves $C$ currently listed in it. We also discuss the determination of the field of definition of the endomorphisms in some special cases.

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arXiv2020

Local opers with two singularities: the case of $\mathfrak{sl}(2)$

We study local opers with two singularities for the case of the Lie algebra sl(2), and discuss their connection with a two-variables extension of the affine Lie algebra. We prove an analogue of the Feigin-Frenkel theorem describing the centre at the critical level, and an analogue of a result by Frenkel and Gaitsgory that characterises the endomorphism rings of Weyl modules in terms of functions on the space of opers.

math.RT↗
arXiv2020

How far is an extension of $p$-adic fields from having a normal integral basis?

Let $L/K$ be a finite Galois extension of $p$-adic fields with group $G$. It is well-known that $\mathcal{O}_L$ contains a free $\mathcal{O}_K[G]$-submodule of finite index. We study the minimal index of such a free submodule, and determine it exactly in several cases, including for any cyclic extension of degree $p$ of $p$-adic fields.

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arXiv2020

Decomposing Jacobians via Galois covers

Let $ϕ:\,X\rightarrow Y$ be a (possibly ramified) cover between two algebraic curves of positive genus. We develop tools that may identify the Prym variety of $ϕ$, up to isogeny, as the Jacobian of a quotient curve $C$ in the Galois closure of the composition of $ϕ$ with a well-chosen map $Y\rightarrow \mathbb{P}^1$. This method allows us to recover all previously obtained descriptions of a Prym variety in terms of a Jacobian that are known to us, besides yielding new applications. We also find algebraic equations for some of these new cases, including one where $X$ has genus $3$, $Y$ has genus $1$ and $ϕ$ is a degree $3$ map totally ramified over $2$ points.

math.AG↗
arXiv2019

Higgs Couplings without the Higgs

The measurement of Higgs couplings constitute an important part of present Standard Model precision tests at colliders. In this article, we show that modifications of Higgs couplings induce energy-growing effects in specific amplitudes involving longitudinally polarized vector bosons, and we initiate a novel program to study these very modifications of Higgs couplings off-shell and at high-energy, rather than on the Higgs resonance. Our analysis suggests that these channels are complementary and, at times, competitive with familiar on-shell measurements; moreover these high-energy probes offer endless opportunities for refinements and improvements.

hep-ph↗
arXiv2019

Explicit Kummer Theory for Elliptic Curves

Let $E$ be an elliptic curve defined over a number field $K$, let $α\in E(K)$ be a point of infinite order, and let $N^{-1}α$ be the set of $N$-division points of $α$ in $E(\bar{K})$. We prove strong effective and uniform results for the degrees of the Kummer extensions $[K(E[N],N^{-1}α) : K(E[N])]$. When $K=\mathbb{Q}$, and under a minimal assumption on $α$, we show that the inequality $[\mathbb{Q}(E[N],N^{-1}α) : \mathbb{Q}(E[N])] \geq cN^2$ holds with a constant $c$ independent of both $E$ and $α$.

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arXiv2018

Computing twists of hyperelliptic curves

We give an efficient algorithm to compute equations of twists of hyperelliptic curves of arbitrary genus over any separable field (of characteristic different from 2), and we explicitly describe some interesting examples.

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arXiv2017

The 1-eigenspace for matrices in $\operatorname{GL}_2(\mathbb{Z}_\ell)$

Fix some prime number $\ell$ and consider an open subgroup $G$ either of $\operatorname{GL}_2(\mathbb{Z}_\ell)$ or of the normalizer of a Cartan subgroup of $\operatorname{GL}_2(\mathbb{Z}_\ell)$. The elements of $G$ act on $(\mathbb{Z}/\ell^n \mathbb{Z})^2$ for every $n\geqslant 1$ and hence also on the direct limit, and we call 1-eigenspace the group of fixed points. We partition $G$ by considering the possible group structures for the 1-eigenspace and show how to evaluate with a finite procedure the Haar measure of all sets in the partition. The results apply to all elliptic curves defined over a number field, where we consider the image of the $\ell$-adic representation and the Galois action on the torsion points of order a power of $\ell$.

math.NT↗
arXiv2017

Reductions of points on algebraic groups

Let $A$ be the product of an abelian variety and a torus defined over a number field $K$. Fix some prime number $\ell$. If $α\in A(K)$ is a point of infinite order, we consider the set of primes $\mathfrak p$ of $K$ such that the reduction $(α\bmod \mathfrak p)$ is well-defined and has order coprime to $\ell$. This set admits a natural density. By refining the method of R.~Jones and J.~Rouse (2010), we can express the density as an $\ell$-adic integral without requiring any assumption. We also prove that the density is always a rational number whose denominator (up to powers of $\ell$) is uniformly bounded in a very strong sense. For elliptic curves, we describe a strategy for computing the density which covers every possible case.

math.NT↗
arXiv2017

The twisting Sato-Tate group of the curve $y^2 = x^{8} - 14x^4 + 1$

We determine the twisting Sato-Tate group of the genus $3$ hyperelliptic curve $y^2 = x^{8} - 14x^4 + 1$ and show that all possible subgroups of the twisting Sato-Tate group arise as the Sato-Tate group of an explicit twist of $y^2 = x^{8} - 14x^4 + 1$. Furthermore, we prove the generalized Sato-Tate conjecture for the Jacobians of all $\mathbb Q$-twists of the curve $y^2 = x^{8} - 14x^4 + 1$.

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